3.1.13
"Double angle" ek angle sum ka special case hi hai: 2 A = A + A sin ( X + Y ) cos ( X + Y ) X = Y = A ek substitution dur hai.
WHY it matters: Ye formulas sin 2 A sin A cos A
HOW we use them: har jagah X = Y = A
sin 2 A = sin ( A + A ) = sin A cos A + cos A sin A
Dono terms ek jaisi hain, toh add ho jaati hain:
Ye step kyun? sin A cos A + cos A sin A 2 sin A cos A
cos 2 A = cos ( A + A ) = cos A cos A − sin A sin A = cos 2 A − sin 2 A
Ab Pythagorean identity == sin 2 A + cos 2 A = 1 ==
sin 2 A = 1 − cos 2 A cos 2 A = cos 2 A − ( 1 − cos 2 A ) = 2 cos 2 A − 1
cos 2 A = 1 − sin 2 A cos 2 A = ( 1 − sin 2 A ) − sin 2 A = 1 − 2 sin 2 A
Intuition TEEN forms kyun?Kaun sa form choose karna hai ye depend karta hai tumhare paas kya hai. Agar problem mein sirf cos A sin A power-reduction rule sin 2 A = 2 1 − cos 2 A
tan 2 A = tan ( A + A ) = 1 − t a n A t a n A t a n A + t a n A
Top: tan A + tan A = 2 tan A tan A ⋅ tan A = tan 2 A
Ye step kyun? Same substitution X = Y = A tan X tan Y tan 2 A Domain note: undefined hoga jab tan 2 A = 1 A = 4 5 ∘ , 13 5 ∘ , … 2 A = 9 0 ∘ tan
Worked example Example 1 —sin A sin 2 A cos 2 A sin A = 5 3 A
Step 1: cos A A cos A = 1 − sin 2 A = 1 − 25 9 = 5 4 Kyun? Pythagoras se dusri side milti hai; acute ⇒ positive root.
Step 2: sin 2 A = 2 sin A cos A = 2 ⋅ 5 3 ⋅ 5 4 = 25 24
Step 3: cos 2 A sin A cos 2 A = 1 − 2 sin 2 A = 1 − 2 ⋅ 25 9 = 25 7 Form 3 kyun? Isse cos A
Worked example Example 2 — Ek identity prove karoProve karo 1 + cos 2 A sin 2 A = tan A
Step 1: Numerator sin 2 A = 2 sin A cos A Step 2: Denominator — woh form choose karo jo 1 + cos 2 A 1 + cos 2 A = 1 + ( 2 cos 2 A − 1 ) = 2 cos 2 A Form 2 kyun? + 1 − 1 Step 3: 2 cos 2 A 2 sin A cos A = cos A sin A = tan A
Worked example Example 3 —tan 2 A tan A = 2 1 tan 2 A Forecast: A ≈ 26. 6 ∘ 2 A ≈ 53. 1 ∘ tan 53. 1 ∘ ≈ 1.33 3 4 Compute: tan 2 A = 1 − ( 2 1 ) 2 2 ⋅ 2 1 = 1 − 4 1 1 = 4 3 1 = 3 4
sin 2 A = 2 sin A Kyun sahi lagta hai: "angle double = sine double" wala pattern linear scaling se match karta hai.
Kyun galat hai: sin linear nahi hai. A = 3 0 ∘ sin 6 0 ∘ = 0.866 2 sin 3 0 ∘ = 1 Fix: sin 2 A = 2 sin A cos A cos A
cos 2 A Kyun sahi lagta hai: teeno forms equal hain, toh "koi bhi" kaam karega.
Kyun bite karta hai: woh value mein equal hain lekin convenience mein nahi. Example 2 mein Form 1 use karne se cos 2 − sin 2 + 1 Fix: form ko data ke saath match karo / us term ke saath jo tum cancel karna chahte ho.
sin A cos A Kyun sahi lagta hai: cos A = 1 − sin 2 A Fix: square root ka ± A A
sin 2 A sin A , cos A 2 sin A cos A
cos 2 A cos 2 A − sin 2 A 2 cos 2 A − 1 1 − 2 sin 2 A
tan 2 A 1 − tan 2 A 2 tan A
cos 2 A cos A 2 cos 2 A − 1
cos 2 A sin A 1 − 2 sin 2 A
sin 2 A = 2 1 − c o s 2 A cos 2 A = 1 − 2 sin 2 A
tan 2 A Jab tan 2 A = 1 A = 4 5 ∘ , 13 5 ∘ , …
Har double-angle formula derive karne ka trick kya hai? Addition formulas mein X = Y = A
1 + cos 2 A 2 cos 2 A
1 − cos 2 A 2 sin 2 A
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho tumhe pata hai do angles ko add karke unka sine aur cosine kaise nikalte hain. "Double angle" bas ek angle ko khud ke saath add karna hai. Toh naye spells sikhne ki zaroorat nahi — jo already pata hain unhe use karo aur same number do baar daalo. Bacha hua tidy karne par shortcut formulas mil jaate hain. cos sin 2 + cos 2 = 1 sin 2 cos 2
Sine: "2 sic " → 2 s in·c os.Cos teen forms: "C an S wap S quares" — cos 2 − sin 2 2 cos 2 − 1 1 − 2 sin 2 1 2 2c²−1 aur 1−2s² .Tan: addition formula jaisa dikhta hai sab kuch doubled/squared ke saath: top 2t, bottom 1−t² .
Compound (Addition) Angle Formulas — woh parent identities jinse ye aate hain.Pythagorean Identity — cos 2 A Power-Reduction / Half-Angle Formulas — cos 2 A Integration of sin²x and cos²x — direct application.Triple Angle Formulas — double + single angle combine karke bante hain.Weierstrass t = tan(A/2) Substitution — tan 2 A
Addition formulas sin cos tan X+Y
Pythagorean identity sin^2+cos^2=1
tan 2A = 2tanA / 1 - tan^2A
Power-reduction sin^2A = 1-cos2A / 2
